In a bowl, whisk together the egg, flour, water, instant yeast, and salt until smooth. Cover and let the batter rest for 30 minutes.
Meanwhile, prepare the beef patties. Combine the ground beef with the dried sage, dried thyme, brown sugar, salt, and garlic powder. Mix until well combined, then divide into 2 equal portions and shape into thin patties.
Cut the ham and cheddar cheese into quarters so they fit comfortably inside the wells of a 4-hole omelet pan.
Heat the pan over medium heat and lightly brush each well with oil. Crack one egg into each well, break the yolk, and gently swirl it with a wooden skewer. Cook for about 1 minute, until the bottoms are slightly set and white while the tops are still runny.
Place the beef patties and ham pieces on top of the eggs. Continue cooking until the eggs are fully set. Carefully flip each egg and cook for another 2 minutes on the meat side.
Remove the egg fillings from the pan and set aside.
Pour enough pancake batter into each well to fill about one-third of the depth. Before the batter begins to set, place one egg filling on top of each portion of batter. Cook for 4–5 minutes, or until the pancake bottoms are set.
To check if they're ready, gently run a wooden skewer or butter knife around the edge. If the pancake spins easily, the bottom and sides are set.
Remove the pancakes from the pan and set aside.
Pour enough pancake batter into each well again, filling them about one-third full. Place the cheddar cheese pieces on top of the batter. Return each pancake to the pan with the meat side facing down so the filling is completely enclosed.
Cook for another 4–5 minutes, until the bottom and sides are set. Again, loosen the edges with a skewer or knife to check for doneness.
Carefully flip the pancake muffins several times until both sides are evenly golden brown and the batter is fully cooked through.
Serve warm, or let the pancake muffins cool completely before wrapping each one tightly in foil and freezing. They freeze up to 2 months.